On 4/20/2013 11:58 AM, WM wrote:> On 20 Apr., 18:49, fom <fomJ...@nyms.net> wrote:>> There are definitions of>> what it means to be inconsistent.>> I know: According to the definitions, an inconsistency is possible but> will never be encountered.

Then it is certainly within your power to formulatedifferent definitions, present them for consideration,and use them to prove an inconsistency.

Until then, an inconsistency is established whenlegitimate proof methods yield proofs for a givenstatement as well as that statement's negation.

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WM is an unabashed ultrafinitist who refuses to fixa largest finite number. Each "n" in his descriptiondepends on the subsequence of triangular numbers.

The triangular numbers correspond withthe number of 'marks' representing numeralsor significant denotations occurring in anyof WM' representations of the form:

12, 13, 2, 1...n, ..., 3, 2, 1...

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This number of 'marks' satisfies a structuralfeature of the natural numbers called adirected set:

Defintion

A binary relation >= in a set D is saidto direct D if and only if D is nonemptyand the following three conditions aresatisfied:

DS1)

If a is an element of D, then a>=a

DS2)

If a, b, c are elements of D suchthat a>=b and b>=c, then a>=c

DS3)

If a and b are elements of D, then thereexists an element c of D such that c>=aand c>=b

So, WM's geometric reasoning for any givenn obtains a finite model domain with itscardinality given by the associatedtriangular number. The triangular numberis the "element c" of condition DS3 fromthe definition.

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Finally, Brouwer's explanation for finitaryreasoning is used because WM refuses tocommit to any mathematical statement withcoherent consistent usage.

Brouwer distinguishes between results withregard to 'endless', 'halted' and'contradictory' in his explanations

"A set is a law on the basis ofwhich, if repeated choices ofarbitrary natural numbers are made,each of these choices eithergenerates a definite sign series,with or without termination of theprocess, or brings about theinhibition of the process togetherwith the definitive annihilationof its result."

WM cannot be an ultrafinitist andexpect others to not hold him totask for it. In constrast toBrouwer, he repeatedly statesthat there is absolutely nocompleted infinity. Therefore,there must be a maximal naturalnumber for his model ofmathematics. Beyond thatnumber, there is no mathematics.

That is WM's belief as surmisedfrom his statements and reasoningsas opposed to what he says withrhetoric.